平衡不完全区组设计方案的序覆盖
On Ordinal Covering of Proposals Using Balanced Incomplete Block Designs
摘要：
在同行审查制度中经常遇到的一个问题是，如何促进由尽可能少的推荐人对给定的一组提案进行两两比较。在[8]中，证明了，如果每个裁判被分配审查k个提案，则ceil{n（n-1）/k（k-1）}裁判是必要的，ceil{n（2n-k）/k^2}裁判足以覆盖所有n（n-1）/2对n个提案。虽然上限保持在下限的2倍之内，但对于小的k值，上限变得相对较大，并且当2 <= k <= n/2时，上限与下限的比率不小于3/2。在本文中，我们证明，如果sqrt（n）<= k <= n/2，那么上界和下界可以更接近，因为它们的比值永远不会超过3/2。这是通过一种新的方法来实现的，该方法使用特定的平衡不完全区组设计族将提案分配给裁判。具体地说，当n/k是素数幂，n整除k^2，并且sqrt（n）<= k <= n/2时，新方法使用ceil{n（n+k）/k^2}个引用。将这个新的上界与[8]中给出的上界进行比较，表明当k趋于sqrt（n）时，新的上界接近下界，而当k趋于n时，[8]中的上界接近下界。因此，当与[8]中的方法结合时，这里给出的新方法提供了一个赋值，当sqrt（n）<= k <= n时，其上限裁判复杂度始终保持在下限的3/2倍之内，从而改进了[8]中描述的赋值。此外，新方法提供了最小覆盖，即，当k = sqrt（n）并且k是素数幂时，它使用可能的最小数量的裁判。
英文摘要：
A frequently encountered problem in peer review systems is to facilitate pairwise comparisons of a given set of proposals by as few as referees as possible. In [8], it was shown that, if each referee is assigned to review k proposals then ceil{n(n-1)/k(k-1)} referees are necessary and ceil{n(2n-k)/k^2} referees are sufficient to cover all n(n-1)/2 pairs of n proposals. While the upper bound remains within a factor of 2 of the lower bound, it becomes relatively large for small values of k and the ratio of the upper bound to the lower bound is not less than 3/2 when 2 <= k <= n/2. In this paper, we show that, if sqrt(n) <= k <= n/2 then the upper and lower bounds can be made closer in that their ratio never exceeds 3/2. This is accomplished by a new method that assigns proposals to referees using a particular family of balanced incomplete block designs. Specifically, the new method uses ceil{n(n+k)/k^2} referees when n/k is a prime power, n divides k^2, and sqrt(n) <= k <= n/2. Comparing this new upper bound to the one given in [8] shows that the new upper bound approaches the lower bound as k tends to sqrt(n) whereas the upper bound in [8] approaches the lower bound as k tends to n. Therefore, the new method given here when combined together with the one in [8] provides an assignment whose upper bound referee complexity always remains within a factor of 3/2 of the lower bound when sqrt(n) <= k <= n, thereby improving upon the assignment described in [8]. Furthermore, the new method provides a minimal covering, i.e., it uses the minimum number of referees possible when k = sqrt(n) and k is a prime power.
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